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Clifford algebra
Clifford algebra is a type of algebra in which the the geometric product is defined. Quick overview The geometric product is one way of generalizing the concept of complex numbers into higher dimensions. The geometric product is a multivector. In three dimensions a multivector is any sum of a scalar, vector, bivector, and a trivector. M = \langle M \rangle_0 + \langle M \rangle_1 + \langle M \rangle_2 + \langle M \rangle_3 Geometric product of vectors A and B = AB = A•B + A∧B :A•B is the dot product of A and B which is a scalar. :A∧B is the wedge product of A and B which is a bivector. In three dimensions there exists a certain unit trivector (e1∧e2∧e3 = e123 = ) whose geometric product with itself is -1. (Multiplying by the equivalent of i converts anything, including itself, to its dual)Some books say to divide by but this only has the effect of changing the sign since 1/ = / 2 = /-1 = - . Therefore in three dimensions this unit trivector is the Clifford algebra equivalent of i. This is easier to understand if we first look at the two-dimensional case. In two dimensions a certain unit bivector would be the equivalent of i. A unit bivector represents a 90-degree turn so the square of a unit bivector would be a 180-degree turn. The geometric product of 2 vectors is a scalar plus a bivector and therefore in two Dimensions it is the clifford algebra equivalent of a complex number. In two dimensions AB = A•B + A∧B = A•B + (A × B) = ||a|| ||b|| ( cos(θ) + sin(θ) ) = re θ :(A × B) is the 2-D cross product of A and B which is a scalar. Terms Clifford algebra in 3 dimensions is called C 3. A bilinear form is a generalization of an inner product. :B(u,v) = scalar A bilinear form is symmetric if the order of the vectors does not matter. :B(u,v) = B(v,u) A bilinear form is Non-degenerate if it is true that whenever u≠0 and v≠0 then B(u,v)≠0 Quadratic form q(x) = B(x,x) where B is a symmetric bilinear form. In a normed space, if the parallelogram lawthe parallelogram law (also called the parallelogram identity) states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals holds, then there is an inner product on V such that \|x\|^2 = \langle x,\ x\rangle for all x \in VWikipedia:Polarization identity Formulas in 3 dimensions e1, e2, and e3 are orthogonal basis unit vectors corresponding to x, y, and z. Geometric product: :e1e2 = e1∧e2 :e1(e1e2) = e2 :e1(e2e3) = e1e2e3 = :(e1e2)(e2e3) = e1e2e2e3 = e1e3 :i = -e2e3 :j = -e3e1 :k = -e1e2 :i2 = j2 = k2 = ijk = -1 : 2 = e1e2e3e1e2e3 = -1 :e1 = e1e1e2e3 = -e1e2e1e3 = e1e2e3e1 = e1 : e1 = e2e3 : 2e1 = e2e3 = -e1 A, B, C, and D are unit vectors :A = a1e1 + a2e2 + a3e3 :B = b1e1 + b2e2 + b3e3 :C = c1e1 + c2e2 + c3e3 :D = d1e1 + d2e2 + d3e3 q(A) = B(A,A) = A•A = (a1)2 + (a2)2 + (a3)2? Wedge product: :3∧5 = 15 :A∧A = 0 :A∧B = bivector (a 2-blade) ::A∧B = (a2b3-a3b2)e23 + (a3b1-a1b3)e31 + (a1b2-a2b1)e12 :A∧B∧C = trivector (a 3-blade) ::A∧B∧C = (a1b2c3 + a2b3c1 + a3b1c2 - a1b3c2 - a2b1c3 - a3b2c1)(e1∧e2∧e3) : A∧B∧C∧D = a three-dimensional quadvector with zero volume and is therefore equal to zero. :(A∧B∧C) • C = A∧B Since A∧A = 0 :0 = (A+B)∧(A+B) :0 = A∧A + A∧B + B∧A + B∧B :0 = 0 + A∧B + B∧A + 0 :0 = A∧B + B∧A Therefore: :-(B∧A) = A∧B This means that rotation from B to A is the negative of rotation from A to B. For Bivectors S and T: :Commutator product = S×T = -(T×S) = ½(ST - TS) :ST = S•T + S×T + S∧T Formulas involving antivectors in 3 dimensions :See also: Grassmann algebra ē1, ē2, and ē3 are pseudo-vectors or anti-vectors: :ē1 = e2∧e3 :ē2 = e3∧e1 :ē3 = e1∧e2 \bar{B} = b_1\bar{e_1} + b_2\bar{e_2} + b_3\bar{e_3} Wedge product of vector and antivector: :(a1e1 + a2e2 + a3e3)∧(b1ē1 + b2ē2 + b3ē3) = (a1b1 + a2b2 + a3b3)(e1∧e2∧e3) = (A•B) A•B = a1b1 + a2b2 + a3b3 AA = A•A The Antiwedge product "∨" operates on antivectors: :ē1∨ē2 = (e2∧e3)∨(e3∧e1) = e3 :ē2∨ē3 = (e3∧e1)∨(e1∧e2) = e1 :ē3∨ē1 = (e1∧e2)∨(e2∧e3) = e2 Use in physics When the electromagnetic field is defined as the multivector sum of an electric field vector and a magnetic field bivector, the four Maxwell equations can be reduced to a single equation. Notes See also *Geometric algebra (A type of Clifford algebra limited to the reals) *symplectic Clifford algebra (Weyl algebras represent the same structure for symplectic bilinear forms that Clifford algebras represent for non-degenerate symmetric bilinear forms)Wikipedia:Weyl algebra References *Electromagnetism using Geometric Algebra versus Components *http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf *http://wiki.c2.com/?CliffordAlgebra *clifford-algebra-a-visual-introduction (uses an asterick to represent the geometric product) *http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/10/GA2015_Lecture2.pdf